Do stallions recognize their own offspring?

Absolutely - probably by scent - and this despite the fact that the mares generally move a distance from the herd to foal.

Stallions will not cover their own daughters, so they remember which foals are theirs even 2-3 years later (colts will, though, cover their own dams...probably because that situation would not happen in the wild, where colts are driven out of their birth herd before puberty). In captivity, stallions have been known to kill foals that they did not sire, and love on and play with ones they did.

So, yes, they seem to know when they are "daddy."